In this article, we show that every bridgeless graph G of order n and maximum degree Δ has an orientation of diameter at most n−Δ+3. We then use this result and the definition NG(H)=⋃v∈V(H)NG(v)∖V(H), for every subgraph H of G, to give better bounds in the case that G contains certain clusters of high‐degree vertices, namely: For every edge e, G has an orientation of diameter at most n−|NG(e)|+4, if e is on a triangle and at most n−|NG(e)|+5, otherwise. Furthermore, for every bridgeless subgraph H of G, there is such an orientation of diameter at most n−|NG(H)|+3. Finally, if G is bipartite, then we show the existence of an orientation of diameter at most 2(|A|− deg G(s))+7, for every partite set A of G and s∈V(G)∖A. This particularly implies that balanced bipartite graphs have an orientation of diameter at most n−2Δ+7. For each bound, we give a polynomial‐time algorithm to construct a corresponding orientation and an infinite family of graphs for which the bound is sharp.
Journal of Graph Theory – Wiley
Published: Jan 1, 2018
Keywords: ; ; ; ; ;
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